c ESO 2010
Astronomy & Astrophysics manuscript no. paper
September 28, 2010
Cosmic-ray driven dynamo
in the interstellar medium of irregular galaxies
H. Siejkowski1 , M. Soida1 , K. Otmianowska-Mazur1 , M. Hanasz2 , and D.J. Bomans3
1
2
arXiv:0909.0926v3 [astro-ph.GA] 27 Sep 2010
3
Astronomical Observatory, Jagiellonian University, ul. Orla 171, 30-244 Kraków, Poland
Toruń Centre for Astronomy, Nicolaus Copernicus University, 87-148 Toruń/Piwnice, Poland
Astronomical Institute of Ruhr-University Bochum, Univeristätsstr. 150/NA7, D-44780 Bochum, Germany
Preprint online version: September 28, 2010
ABSTRACT
Context. Irregular galaxies are usually smaller and less massive than their spiral, S0, and elliptical counterparts. Radio observations
indicate that a magnetic field is present in irregular galaxies whose value is similar to that in spiral galaxies. However, the conditions
in the interstellar medium of an irregular galaxy are unfavorable for amplification of the magnetic field because of the slow rotation
and low shearing rate.
Aims. We investigate the cosmic-ray driven dynamo in the interstellar medium of an irregular galaxy. We study its efficiency under
the conditions of slow rotation and weak shear. The star formation is also taken into account in our model and is parametrized by the
frequency of explosions and modulations of activity.
Methods. The numerical model includes a magnetohydrodynamical dynamo driven by cosmic rays that is injected into the interstellar
medium by randomly exploding supernovae. In the model, we also include essential elements such as vertical gravity of the disk,
differential rotation approximated by the shearing box, and resistivity leading to magnetic reconnection.
Results. We find that even slow galactic rotation with a low shearing rate amplifies the magnetic field, and that rapid rotation with
a low value of the shear enhances the efficiency of the dynamo. Our simulations have shown that a high amount of magnetic energy
leaves the simulation box becoming an efficient source of intergalactic magnetic fields.
Key words. MHD - ISM: magnetic fields - Galaxies: irregular - Methods: numerical
1. Introduction
Irregular galaxies have lower masses than typical spirals and ellipticals. In addition, they have irregular distributions of the starforming regions, and rotations that are slower than spiral galaxies by half an order of magnitude (Gallagher & Hunter 1984).
The rotation curves of irregular galaxies are non-uniform and
have a weak shear.
Radio observations of magnetic fields in spiral galaxies indicate that their magnetic fields have strong ordered (1–5 µG)
and random (9–15 µG) components (Beck 2005). A plausible
process fueling the growth in the magnetic energy and flux
of these galaxies is magnetohydrodynamical dynamo (Widrow
2002; Gressel et al. 2008). The vital conditions required for the
dynamo to effectively amplify the magnetic field are rapid rotation and shear. In irregular galaxies, both quantities seem to be
too low to initiate efficient dynamo action. In contrast, the observations of magnetic field in irregular galaxies indicate that
these galaxies could have strong and ordered magnetic fields
(e.g., Chyży et al. 2000, 2003; Kepley et al. 2007; Lisenfeld et
al. 2004).
The most spectacular radio observations to date of irregulars were those performed for the galaxy NGC 4449 (Chyży et
al. 2000). The total strength of its magnetic field is about 14 µG
with a ordered component reaching locally values of 8 µG. These
are similar to the intensities observed for large spirals. A high
number of H ii regions and slow rotation is also observed with
quite large velocity shear (Valdez-Gutiérrez et al. 2002). The radio observations of H i around the galaxy indicate that this object
is embedded in two large H i systems that counter-rotate with re-
spect to the optical part of this galaxy (Bajaja et al. 1994; Hunter
et al. 1998, 1999). In addition to these H i clouds, NGC 4449
contains an unusual ring of H i in the outer part of the optical
disk (Hunter et al. 1999). This complicated topology of the H i
velocity field could help in achieving efficient magnetic field amplification (see Otmianowska-Mazur et al. 2000).
Chyży et al. (2003) found that two other irregular galaxies,
NGC 6822 and IC 10 are also magnetized. The former has a very
low total magnetic field weaker than 5 µG, a small number of
H ii regions, and almost rigid rotation (see Sect. 2). These properties are directly related to the efficiency of the dynamo process in galaxies (see Otmianowska-Mazur et al. 2000; Hanasz et
al. (2006, 2009) that weakly amplifies the magnetic field in this
galaxy. The irregular IC 10 has a total magnetic field strength
that varies between 5 and 15 µG with no ordered component.
Observations performed by Chyży et al. (2003) indicate that the
total magnetic field is correlated with the number of H ii regions.
The number of the regions is higher than in NGC 6822, and the
rotation of IC 10 has a partly differential character (see Sect. 2).
Both conditions lead to more rapid magnetic field amplification
than for NGC 6822. As for NGC 4449, IC 10 is embedded in
a large cloud of H i, which counter-rotates with respect to the
inner disk (Wilcots & Miller 1998).
Klein et al. (1993) inferred that the Large Magellanic Cloud
(LMC) has a large-scale magnetic field that has the shape of
a trailing spiral structure, similar to normal spiral galaxies. It is
possible that the amplification of the magnetic field is connected
to the differential rotation of this galaxy present beyond a cer-
2
H. Siejkowski et al.: CR driven dynamo in the ISM of irregular galaxies
Table 1. Main properties of NGC 4449, NGC 6822, and IC 10
Type
Distance [Mpc]
Diameter [kpc]
SFR
< Btot > [µG]
< Breg > [µG]
NGC 4449
IBm
4.21
5.73
high
5–15
≃8
NGC 6822
IB(s)m
0.50
1.71
low
≤5
≤3
IC 10
dIrr
0.66
1.27
high
5–15
≤3
In the following rows we present: the morphological type of the object
(LEDA), the distance (Karachentsev et al. 2004), the size calculated
from D25 (LEDA) and distance, the star formation rate (SFR), and the
results of the radio analysis of Chyży et al. (2000, 2003).
NGC 6822, and IC 10 acquired by Chyży et al. (2000, 2003).
The main properties of these objects are presented in Table 1.
From the rotation curves of IC 10 (Fig. 1, top panel, solid
line) obtained by Wilcots & Miller 1998, NGC 6822 (Fig. 1,
top panel, dashed line) obtained by Weldrake et al. 2003, and
NGC 4449 (Fig. 1, top panel, dot-dashed line) by ValdezGutiérrez et al. 2002, we computed the angular velocity and
shearing rate of each galaxy (see Sect. 5.1). For the two first
galaxies, we use H i data. In the case of NGC 4449, we restricted
our analysis to the internal region and used Hα data, because of
its very complex velocity pattern.
In the velocity pattern of the IC10, we can see a central part
with a solid-body rotation, which flattens to a constant value
vrot ≃ 30 km s−1 at r = 352 pc. The NGC 6822 rotation curve is
a monotonically increasing function with a square root slope and
the highest value vrot ≃ 60 km s−1 at r = 5.7 kpc. The rotation
curve of NGC 4449 is highly disturbed and reaches a maximum
value of vrot ≃ 40 km s−1 at radius of 2 kpc.
tain radius (Klein et al. 1993; Luks & Rohlfs 1992; Gaensler et
al. 2005).
We note that polarized radio emission is detected in the irregular galaxy NGC 1569. This galaxy has a very high star formation rate (Martin 1998) and exhibits bursts of activity in its 3. Description of the model
past (Vallenari & Bomans 1996). The radio observation of this The CR-driven dynamo model consists of the following elegalaxy by Lisenfeld et al. (2004) found that the galaxy has large- ments (based on Hanasz et al. 2004, 2006):
scale magnetic fields in the disk and halo. Furthermore, they
found that their data agree that a convective wind could allow (1) The cosmic ray component is a relativistic gas described by
for escape of cosmic-ray electrons in to the halo. These observaa diffusion-advection transport equation. Typical values of
tions are the main reason for undertaking our CR-driven dynamo
the diffusion coefficient are (3 ÷5) ×1028 cm2 s−1 (see Strong
calculations in irregular galaxies. In addition to Lisenfeld et al.
et al. 2007) at energies of around 1 GeV, although in our
(2004), the radio observations of Kepley et al. (2007) showed
simulations we use reduced values (see Sect. 4.1).
that the large-scale magnetic arms visible in NGC 1569 are (2) Anisotropic diffusion of CR. Following Giacalone & Jokipii
(1999) and Jokipii (1999), we assume that the CR gas difaligned perpendicularly to the disk and that the northern part of
the disk of the galaxy is inclined at a different angle.
Kronberg et al. (1999) realized that dwarf galaxies (apart
from their low masses) could serve as an efficient source of gas
and magnetic fields in the intergalactic medium (IGM) during
their initial bursts of star formation in the early Universe. In the
case of star-forming dwarf galaxies, we expect that the dominant
driver of a galactic wind are cosmic rays, in contrast to large spirals, such as the Milky Way, where the thermal driving is most
significant (Everett et al. 2008). Therefore, we applied the model
of the CR-driven dynamo to the interstellar medium and conditions of an irregular galaxy and try to find how much of the
magnetic energy can be expelled from the dwarf galaxies to the
IGM.
Many questions about magnetic field amplification in irregular galaxies remain unresolved. The physical explanation of
this process is difficult to establish because these galaxies rotate
slowly, almost like a solid body. In this paper, we check how our
model of cosmic-ray driven dynamo, which effectively describes
spiral galaxies (Hanasz et al. 2004, 2006, 2009), can be applied
to irregular galaxies. In the present numerical experiment we attempt to answer how the model input parameters observed in irregulars (small gravitational potential, gas density, low rotation,
and small shear) influence the magnetic fields within them. We
have not taken into account the magnetic field possibly injected
by stars. We plan to study this in the future. We found that in
certain conditions achievable for irregulars it is possible to have
efficient magnetic field amplification.
Fig. 1. Observational rotation characteristics of IC 10,
NGC 6822, and NGC 4449. We present, from top to bottom: the rotation curves (references in Sect. 2), the calculated
2. Observations of irregular galaxies
angular velocity, and the computed shear parameter q (for
To study properties of the irregulars and determine the input pa- details see Sect. 5.1) respectively. The shaded region marks the
rameters for our simulations, we use observations of NGC 4449, range of parameters presented in this paper.
H. Siejkowski et al.: CR driven dynamo in the ISM of irregular galaxies
(3)
(4)
(5)
(6)
3
fuses anisotropically along magnetic field lines. The ratio of
the perpendicular to parallel CR diffusion coefficients suggested by the authors is 5%.
Localized sources of CR. In the model, we apply the random
explosions of supernovae in the disk volume. Each explosion is a localized source of cosmic rays. The cosmic ray
input of individual SN remnant is 10% of the canonical kinetic energy output (1051 erg) and distributed over several
subsequent time steps.
Resistivity of the ISM to enable the dissipation of the smallscale magnetic fields (see Hanasz et al. 2002 and Hanasz &
Lesch 2003). In the model, we apply the uniform resistivity
and neglect the Ohmic heating of gas by the resistive dissipation of magnetic fields.
Shearing boundary conditions and tidal forces following the
prescription by Hawley, Gammie & Balbus (1995), are implemented to reproduce the differentially rotating disk in the
local approximation.
Realistic vertical disk gravity following the model by
Ferrière (1998) modified by reducing the contribution of disk
and halo masses by one order of the magnitude, to adjust the
irregular galaxy environment.
We apply the following set of resistive MHD equations:
∂ρ
+ ∇ · (ρV) = 0,
∂t
∂e
+ ∇ · (eV) = −p(∇ · V),
∂t
!
∂V
1
B2
+ (V · ∇)V = − ∇ p + pcr +
∂t
ρ
8π
B · ∇B
− 2Ω × V + 2qΩ2 xê x ,
+
4πρ
∂B
= ∇ × (V × B) + η△B,
∂t
p = (γ − 1)e, γ = 5/3,
(1)
(2)
(3)
(4)
(5)
where q = −d ln Ω/d ln R is the shearing rate, R is a galactocentric radius, η represents the ISM resistivity, γ is the adiabatic index of thermal gas, pcr is the cosmic-ray pressure, and the other
symbols have their usual meaning. In the equation of motion, the
term ∇pcr is included (see Berezinskii et al. 1990). The thermal
gas is approximated by an adiabatic medium.
The cosmic ray component is an additional fluid described
by the diffusion-advection equation (see e.g., Schlickeiser &
Lerche 1985)
∂ecr
+ ∇(ecr V) = ∇(K̂∇ecr ) − pcr (∇ · V) + QSN ,
(6)
∂t
where QSN is the source term of the cosmic-ray energy density injected locally from randomly exploding SN remnants. The
cosmic-ray fluid is described by an adiabatic equation of state
with adiabatic index γcr :
pcr = (γcr − 1)ecr , γcr = 14/9.
(7)
The K̂ is an diffusion tensor described by the formula:
Ki j = K⊥ δi j + (Kk − K⊥ )ni n j , ni = Bi /B,
(8)
adopted following the argumentation of Ryu et al. (2003).
The vertical gravitational acceleration is taken from Ferrière
(1998). We reduced both contributions of disk and halo by a factor of 10, the scale length of the exponential disk to LD = 2 kpc,
Fig. 2. Example slices of a domain taken from simulation
R.01Q1 at t = 660 Myr. On the slices, the Parker loop is produced by cosmic rays from supernovae explosions.
and scale length of halo to LH = 1 kpc. In our computations, we
incorporated the formula :
!
2
2
Z
−10
−2 R∗ + LH
−gz (R, Z) = (1.7 · 10 cm s )
R + L2H 1 kpc
!
R − R∗
Z
, (9)
+ (4.4 · 10−10 cm s−2 ) exp −
p
LD
Z 2 + (0.2 kpc)2
where R∗ is the distance of the origin of the simulation box from
the galactic center and Z is the height above the galactic mid
plane.
4. Model setup and parameters
4.1. Model setup
The 3D cartesian domain size is 0.5 kpc × 1 kpc × 8 kpc in x, y, z
coordinates corresponding to the radial, azimuthal, and vertical
directions, respectively. The grid size is 20 pc in each direction.
The boundary conditions are sheared-periodic in x, periodic in
y, and an outflow in z direction. The domain is placed at the
galactocentric radius R∗ = 2 kpc. In Fig. 2, we present example
slices through the simulation domain. The left panel shows the
CR energy density with the magnetic field vectors and the right
panel shows the gas density with velocity vectors.
The positions of SNe are chosen randomly with a uniform
distribution in the xy plane and a Gaussian distribution in the vertical direction. The scaleheight of SN explosions in the vertical
4
H. Siejkowski et al.: CR driven dynamo in the ISM of irregular galaxies
Table 2. List of models
Model
R.01Q1a
R.02Q1
R.03Q1b
R.04Q1
R.05Q1c
R.01Q0
R.01Q.5
R.01Q1a
R.01R1.5
R.05Q0
R.05Q.5
R.05Q1c
R.05R1.5
SF3R.03Q.5
SF3R.03Q1
SF10R.03Q.5d
SF10R.03Q1b
SF30R.03Q.5
SF30R.03Q1
M10/100
M20/200d
M50/100
M100/200
M100/100
FIRST
Ω
[Myr−1 ]
0.01
0.02
0.03
0.04
0.05
0.01
0.01
0.01
0.01
0.05
0.05
0.05
0.05
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
q
1
1
1
1
1
0
0.5
1
1.5
0
0.5
1
1.5
0.5
1
0.5
1
0.5
1
0.5
0.5
0.5
0.5
0.5
0.5
f
[kpc−2 Myr−1 ]
10
10
10
10
10
10
10
10
10
10
10
10
10
3
3
10
10
30
30
10
10
10
10
10
2.5
Tp
[Myr]
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
100
200
100
100
100
2000
Ta
[Myr]
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
10
20
50
200
100
50
Subsequent columns show: the model name, the angular velocity Ω,
the shearing parameter q, the frequency of SN explosions f , the period
of SNe modulation T p , and the duration of SNe activity in one the
period T a . See Sect. 4.2 for details. The horizontal lines distinguish
between different simulation series. Models with the same superscript
(a , b , c , and d ) point to the same experiments, but are written for clarity.
direction is 100 pc, and the CR energy that originates in an explosion is injected instantaneously into the ISM with a Gaussian
radial profile (rSN = 50 pc). In addition, the SNe activity is modulated during the simulation time by a period T p and an activity
time T a .
The applied value of the perpendicular CR diffusion coefficient is K⊥ = 103 pc2 Myr−1 = 3 × 1026 cm2 s−1 and the parallel
one is Kk = 104 pc2 Myr−1 = 3 × 1027 cm2 s−1 . The diffusion
coefficients are 10% of realistic values because of the simulation
timestep, which becomes prohibitively short when the diffusion
is too high.
The initial state of the system represents the magnetohydrostatic equilibrium with the horizontal, purely azimuthal magnetic field with pmag /pgas = 10−4 , which corresponds to the
mean value of magnetic field in the simulation box of 5 nG.
Magnetic diffusivity η is set to be 100 pc2 Myr−1 , which corresponds to 3×1025 cm2 s−1 in cgs units (Lesch 1993). The column
density of gas is ̺gas = 6 × 1020 cm−2 (taken from observations,
see Gallagher & Hunter 1984) and the initial value of the isothermal sound speed is set to be ciso = 7 km s−1 .
4.2. Model parameters
We present the results of four simulation series corresponding to
different sets of the CR-dynamo parameters. Details of all computed models are shown in Table 2. The model name consists
of a combination of four letters: R, Q, SF and M followed by
a number. The letter R means the angular velocity (rotation), Q is
the shearing rate, SF is the supernova explosion frequency and
M represents for its modulation during the simulation time, and
the numbers determine the value of the corresponding quantity.
Only the modulation symbol is followed by two numbers, the
first corresponding to the time of the SNe activity and the second to a period of modulation. Values of the parameters are given
in the following units: angular velocity in Myr−1 , supernova explosion frequency in kpc−2 Myr−1 , and the modulation times in
Myr. For example, a model named ”R.01Q1” denotes a simulation where Ω = 0.01 Myr−1 and q = 1, and the name ”M50/100”
denotes an experiment with T a = 50 Myr and T p = 100 Myr. The
last model in Table 2, named FIRST, points to an experiment, in
which only during the first 50 Myr supernovae are active and
after that time CR injection stops.
5. Results
5.1. Shear parameter q obtained from observations
The shearing rate parameter q (defined in Sect. 3) is calculated
numerically from the observational rotation curves using a second order method
!
1 vi+1 − 2vi + vi−1
Ri vi+1 − vi−1
,
(10)
+
qi = 1 −
vi Ri+1 − Ri−1 2 Ri+1 − Ri−1
applied to the radial velocities vi measured at Ri of the observed
rotation curve. Calculations are performed only where the rotation curve is smooth enough, because of the enormous velocity
fluctuations and low spatial resolution, which cause large dispersions in our results. The estimated shearing parameters from
observational rotation curves are presented in Fig. 1. Different
values of the parameter q, correspond to the following interpretations: when q < 0, the rotation velocity increases faster than
a solid body; when q = 0, we have solid body rotation; when
0 < q < 1, the rotation velocity increases slower than a solid
body; q = 1 relates to a flat rotation curve; for q > 1, the azimuthal rotation decreases with R. We found that the shearing
rates are high in all three galaxies and due to the large variations
in the rotation curves, q changes rapidly. However, in the case
of NGC 6822, q gradually increases from 0 to 1.5 with galactocentric distance. For the galaxies IC 10 and NGC 4449, the
calculated local shearing rates vary from −1.5 to 3. This scatter
in the results is caused by the large fluctuations in the measured
rotation velocities.
5.2. The magnetic field evolution
We study dependence of the magnetic field amplification on the
parameters describing the rotation curve, namely, the shearing
rate q and the angular velocity Ω. The evolution in the total
magnetic field energy E B and total azimuthal flux Bφ for different values of Ω is shown in Fig. 3, left and right panel, respectively. Models with higher angular velocities, starting from
0.03 Myr−1 , initially exhibit exponential E B growth and after
about 1 200 Myr, the process saturates (see Sect. 6 for the discussion). The saturation values of magnetic energy for these three
models are similar and E B exceeds the value 104 in the normalized units. The magnetic energy in the models R.01Q1 and
R.02Q1 grows exponentially during the whole simulation and
does not reach the saturation level. The final E B for R.02Q1 is
around 4 × 103 and for the slowest rotation (R.01Q1) in our
sample is only 20. The total azimuthal magnetic flux evolution (Fig. 3, right) shows that a higher angular velocity produces a higher amplification. The azimuthal flux for models with
H. Siejkowski et al.: CR driven dynamo in the ISM of irregular galaxies
5
Table 3. Summary of the simulations results
Model
log Ē end
B
log E out
B
end
E out
B / Ē B
R.01Q1a
R.02Q1
R.03Q1b
R.04Q1
R.05Q1c
R.01Q0
R.01Q.5
R.01Q1a
R.01Q1.5
R.05Q0
R.05Q.5
R.05Q1c
R.05Q1.5
SF3R.03Q.5
SF3R.03Q1
SF10R.03Q.5d
SF10R.03Q1b
SF30R.03Q.5
SF30R.03Q1
M10/100
M20/200d
M50/100
M100/200
M100/100
FIRST
1.37
3.55
4.03
4.17
3.95
-1.73
0.06
1.37
1.26
-1.00
4.03
3.95
3.81
3.14
3.28
3.87
4.03
2.79
3.30
3.78
3.87
4.05
4.11
3.86
2.37
0.07
2.02
2.89
3.29
3.43
-0.42
-0.35
0.07
0.15
0.04
3.38
3.43
3.07
2.23
2.28
2.28
2.89
2.77
3.05
2.73
2.28
2.92
2.67
2.72
1.33
0.05
0.03
0.07
0.13
0.30
20.48
0.39
0.05
0.08
10.78
0.22
0.30
0.18
0.12
0.10
0.03
0.07
0.96
0.57
0.09
0.03
0.07
0.04
0.07
0.09
hBi
[µG]
0.068
0.835
1.206
1.285
1.120
0.001
0.011
0.068
0.056
0.003
1.168
1.120
0.934
0.190
0.243
0.991
1.206
0.190
0.243
0.833
0.243
1.001
1.352
0.751
0.113
te
[Myr]
1 219
440
375
346
509
–
–
1 219
1 232
–
363
509
375
506
444
403
375
629
547
422
403
404
393
422
572
TΩ
[Myr]
628
314
209
157
125
628
628
628
628
125
125
125
125
209
209
209
209
209
209
209
209
209
209
209
209
Description
slow rotation
slow rotation
medium rotation
fast rotation
fast rotation
low shear
medium shear
medium shear
high shear
low shear
medium shear
medium shear
high shear
low SFR
low SFR
medium SFR
medium SFR
high SFR
high SFR
constantly exploding SNe
SN activity during first 50Myr
The subsequent columns show: the model name, the mean value of total magnetic energy Ē end
B over past 50 Myr, the total outflow of magnetic
energy during whole simulation (see Eq. 12), the ratio of these two quantities, the final mean value of the magnetic field in a disc midplane hBi
(|z| < 20 pc), the e-folding time of magnetic flux increase te , galaxy revolution timescale T Ω , and a short description of a model. Superscripts are
explained in Table 2. Values of magnetic field energy are normalized to the initial value.
Ω ≥ 0.02 Myr−1 exceeds the value 102 . Model R.01Q1 does not
enhance the azimuthal flux at all.
In Fig. 4, we present results for models with different shearing rate values. The evolution of E B and Bφ in models R.05Q.5
and R.05Q1.5 follows the evolution of model R.05Q1, which is
described in the previous paragraph. Similar behavior is noted
for R.01Q1.5 and R.01Q1, but the model R.01Q.5 alone sustains
its initial magnetic field. In the case of models with no shear
(R.01Q0, R.05Q0), the initial magnetic field decays.
We check how the frequency and modulation of SNe influence the amplification of magnetic fields. The evolution in total
magnetic field energy and total azimuthal flux for different supernova explosion frequencies are shown in Fig. 5, left and right
respectively. The total magnetic energy evolution for all models is similar, but in the case of the azimuthal flux we observe
differences between the models. The most efficient amplification of Bφ appears for SF10R.03Q.5 and SF10R.03Q1, and for
other models the process is less efficient. In addition, for models SF30R.03Q.5 and SF30R.03Q1, we observe a turnover in
magnetic field direction. The results suggest that the dynamo
requires higher frequencies of supernova explosions to create
more regular fields, although, if the explosions occur too frequently, this process is suppressed because of the overlapping
turbulence. The analysis of the M models (Fig. 6) shows that
the dynamo process depends on the duration of the phase when
supernova activity switches off. The fastest growth of magnetic
field amplification occurs for models M100/200 and M50/100 in
which periods of SN activity occupy half of the total modulation
period. The amplification is apparently weaker in cases of short
SN activity periods (M10/100, M20/200) and continuous activ-
ity (M100/100). In all M models, the final E B reaches a value
of the order of 104 . For Bφ evolution, we found that the magnetic flux in the model M50/100 increases exponentially and saturates after 1 300 Myr. Similar behavior is exhibited by the models M10/100 and M100/100 but the saturation times occur after
1 700 Myr and the growth is slower than in the previous case.
The model M100/200 after exponential growth at t = 1 650 Myr
probably begins to saturate, but to quantify this exactly the simulation should continue. The model M20/200 grows exponentially
and does not appear to saturate.
In the case of the model FIRST (Fig. 6), we found that after about 8 galaxy revolutions the growth in E B and Bφ stops.
The total magnetic field energy increase exponentially and after reaching a maximum at t = 1 400 Myr, it exceeds the value
2 ×102, whereas the azimuthal flux saturates after 1 600 Myr and
afterwards starts to decay gently.
5.3. Magnetic field outflow
To measure the total production rate of magnetic field energy
during the simulation time, we calculate the outflowing E out
B
through the xy top and bottom domain boundaries. To estimate
the magnetic energy loss, we compute the vertical component of
the Poynting vector
S z = (B xv x + Byvy )Bz − (B2x + B2y )vz .
(11)
6
H. Siejkowski et al.: CR driven dynamo in the ISM of irregular galaxies
Fig. 3. Evolution of the total magnetic energy E B (left panel) and the total azimuthal flux Bφ (right) for models with different rotation.
Both quantities are normalized to the initial value.
Fig. 4. Evolution of the total magnetic energy E B (left panel) and the total azimuthal flux Bφ (right) for models with different
shearing rate and rotation. Both quantities are normalized to the initial value. For models with Ω = 0.01 Myr−1 , the Bφ values has
been multiply by factor 20 – proper y-axis for these plots is on the right side of the frame.
This value is computed in every cell belonging to the top and
bottom boundary planes and then integrated over the entire area
and time:



1 X X
out
t
t
 (S z )i jkmin − (S z )i jkmax  ∆t,
EB =
(12)
∆z t
ij
where kmin and kmax refer to the bottom and top boundary respectively, ∆z is the vertical dimension of single cell, t is the simulation time, and ∆t is the timestep. For models with a low dynamo
efficiency most of the initial magnetic field energy is transported
out of the simulation box. In some cases (i.e., all models except R.01Q0 and R.05Q0), we find that the energy loss E out
B is
comparable to the energy remaining inside the domain Ē end
B . In
end
these models, the ratio E out
/
Ē
varies
from
0.03
to
0.96
and
B
B
is highly dependent on the supernova explosion frequency. For
models with q = 0, in which the dynamo does not operate, the
outflowing energy originate only from the initial condition for
the magnetic field. The results show that the outflowing magnetic energy is substantial (see Table 3) suggesting, that irregular
galaxies because of their weaker gravity can be efficient sources
of intergalactic magnetic fields.
6. Discussion
The most effective magnetic field amplification that we have
found is that in the model R.04Q1, which we associate with the
galaxy NGC 4449. This galaxy has the highest star formation
rate in our sample of three irregulars. The rotation is rapid, reaching 40 km/s, and, for a wide range of radii, the shear is strong.
The numerical model predicts an effective magnetic field amplification and NGC 4449 indeed hosts the strongest magnetic
H. Siejkowski et al.: CR driven dynamo in the ISM of irregular galaxies
7
Fig. 5. Evolution of the total magnetic energy E B (left panel) and the total azimuthal flux Bφ (right) for models with different
supernova explosion frequency and shearing rate. Both quantities are normalized to the initial value.
Fig. 6. Evolution of the total magnetic energy E B (left panel) and the total azimuthal flux Bφ (right) for models with different times
of supernovae modulation. Both quantities are normalized to the initial value.
field among the irregulars, both in terms of its total and ordered
component of 14 µG and 8 µG, respectively (Chyży et al. 2000).
has a relatively low mass and its shallow gravitational potential
makes the escape of its magnetized ISM easier.
The next galaxy IC 10 forms stars at a lower rate than
NGC 4449. The shear is strong and it is a rapid rotator. We can
compare this galaxy to our model R.05Q1, where we see the
fastest initial growth of the total magnetic energy, but the final
value is smaller than that in the case of slower rotation. The total azimuthal flux evolves in a complex way with a reversal in
the mean magnetic field direction. This may indicate that because of its relatively rapid rotation and small size, instabilities
can evolve faster. Separate instability domains can mix (overlap) with each other resulting in a chaotic though still amplified
magnetic field. Consequently IC 10 exhibits a strong total magnetic field of 5–15 µG (as estimated by Chyży et al. 2003). We
notice that by increasing the rotation speed, the amount of magnetic energy expelled from the galaxy grows (see Table 3). IC 10
NGC 6822 forms stars at the slowest rate in our sample. It is
also the slowest rotator. The rotation is almost rigid in its central part (out to ∼0.5 kpc) gradual becoming differential at larger
galactocentric distances but the calculated shearing rate remains
small. We can explain its weak magnetic field of lower than 5 µG
(Chyży et al. 2003) by comparing with our model FIRST: a single burst of star-forming activity in the past followed by a long
(lasting until present) period of almost no star-forming activity.
In this model, the magnetic field, amplified initially, fades since
the star formation stops. This star-forming activity was analyzed
for spiral galaxies by Hanasz et al. (2006), who measured a linear growth in the magnetic field. We can explain this by using
a shorter simulation time (by about a factor of two) than in our
case, but it may indicate that in irregulars the magnetic field is
more easily expelled from the galaxy.
8
H. Siejkowski et al.: CR driven dynamo in the ISM of irregular galaxies
Our models, for which we measure amplification in E B and
Bφ during the simulation, produce a mean magnetic field of order
1–0.5 µG (Table 3) within a disc volume. Models with slower
growth of magnetic field reach values of hBi around tens of nG,
and models with no dynamo action diffuse the initial magnetic
field outside the simulation box.
In Table 3, we present the average e-folding time of the magnetic flux increase te and the galactic revolution period T Ω . The te
of most models is in between 300 and 600 Myr. For spiral galaxies, Hanasz et al. (2006, 2009) found that the e-folding timescale
is about 150–190 Myr. The difference between spirals and irregulars is probably caused by rotation, which is much more rapid
in spirals.
In most of our models, large fractions of the magnetic field
are expelled out of the computational domain – almost 20%–
30% of the magnetic energy maintained in the galaxy. In general more rapid rotation and a high SNe rate make it easier for
the magnetized medium to escape. However, for higher shear
rate, the share of the expelled magnetic field is lower. The
optimal set of parameters, from this point of view, is represented by the model R.05Q1, which we relate to IC 10. In the
other two galaxies, the expelled field is also high – about 10%.
Models with excessive star formation increase this fraction to
60% (SF30R.03Q.5) or even 96% (SF30R.03Q1). Therefore, the
irregular galaxies, in particular compact and intensively forming
stars such as IC 10, are an important source of magnetic field in
intergalactic and intracluster media, as predicted by Kronberg et
al. (1999).
For most of our models we found that the value of the magnetic field strength in the vicinity of a galaxy (at z = 4 kpc)
is about 30-200 nG. Only models with low magnetic-field production rates produce negligible magnetic fields at this height.
This area is the highest point in our simulation domain above
the galactic midplane and can be considered as a transition region between the ISM and the IGM. Hence, the magnetic field
strengths in the models can be an upper limit to the values in the
IGM region. Our estimates are in an agreement with previous
studies, including Ryu et al. (1998), who demonstrated that in
largescale filaments, magnetic fields of about 1 µG may exists,
Kronberg et al. (1999), who calculated that on Mpc scales the average magnetic field strength is about 5 nG, and Gopal-Krishna
& Wiita (2001), who showed that radio galaxies can seed the
IGM with a magnetic field of the order 10 nG during the quasar
era. However, to obtain realistic profile or even the maximum
possible range of expelled magnetic field in the case of dwarf
galaxies we should take into account the interaction between the
IGM and ISM (pressure), which is not included in our model.
We plan to extend our research in this respect in future work.
7. Conclusions
We have described the evolution in the magnetic fields of irregular galaxies in terms of a cosmic-ray driven dynamo. Our
cosmic-ray driven dynamo model consists of (1) randomly exploding supernovae that supply the CR density energy, (2) shearing motions due to differential rotation, and (3) ISM resistivity.
We have studied the amplification of magnetic fields under different conditions characterized by the rotation curve (the angular velocity and the shear) and the supernovae activity (its frequency and modulation) typical of irregular galaxies. We have
found that:
– in the presence of slow rotation and weak shear in irregular
galaxies, the amplification of the total magnetic field energy
is still possible;
– shear is necessary for magnetic field amplification, but the
amplification itself depends weakly on the shearing rate;
– higher angular velocity enables a higher efficiency in the CRdriven dynamo process;
– the efficiency of the dynamo process increases with SNe activity, but excessive SNe activity reduces the amplification;
– a shorter period of halted SNe activity leads to faster growth
and an earlier saturation time in the evolution of azimuthal
magnetic flux;
– for high SNe activity and rapid rotation, the azimuthal flux
reverses its direction because of turbulence overlapping;
– because of the shallow gravitation potential of an irregular
galaxy, the outflow of magnetic field from the disk is high,
suggesting that they may magnetize the intergalactic medium
as predicted by Kronberg et al. (1999) and Bertone et al.
(2006).
The performed simulations indicate that the CR-driven dynamo can explain the observed magnetic fields in irregular
galaxies. In future work we plan to determine the influence of
other ISM parameters and perform more global simulations of
these galaxies.
Acknowledgements. This work was supported by Polish Ministry of
Science and Higher Education through grants: 92/N-ASTROSIM/2008/0 and
3033/B/H03/2008/35. Presented computations have been performed on the
GALERA supercomputer in TASK Academic Computer Centre in Gdańsk.
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